If F(x) = int_{0}^{x} frac{cos t}{1+t^{2}} dt | vedclass

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(D) Given the function $F(x) = \int_{0}^{x} \frac{\cos t}{1+t^{2}} dt$ for $0 \leq x \leq 2\pi$.To determine the intervals of increase and decrease, w

Vedclass · Accepted Answer

$F$ is increasing in $(0, \frac{\pi}{2})$ and $(\frac{3\pi}{2}, 2\pi)$ and decreasing in $(\frac{\pi}{2}, \frac{3\pi}{2})$

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(D) Given the function $F(x) = \int_{0}^{x} \frac{\cos t}{1+t^{2}} dt$ for $0 \leq x \leq 2\pi$.To determine the intervals of increase and decrease, we find the derivative $F'(x)$ using the Leibniz rule:$F'(x) = \frac{\cos x}{1+x^{2}}$.Since $1+x^{2} > 0$ for all $x$, the sign of $F'(x)$ depends solely on $\cos x$.$F(x)$ is increasing when $F'(x) > 0$, which occurs when $\cos x > 0$. In the interval $[0, 2\pi]$, $\cos x > 0$ for $x \in (0, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi)$.$F(x)$ is decreasing when $F'(x) Thus, $F$ is increasing in $(0, \frac{\pi}{2})$ and $(\frac{3\pi}{2}, 2\pi)$ and decreasing in $(\frac{\pi}{2}, \frac{3\pi}{2})$.

If $F(x) = \int_{0}^{x} \frac{\cos t}{1+t^{2}} dt$, where $0 \leq x \leq 2\pi$, then which of the following is true?

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